Fuzzy sets as one of the elements of artificial intelligence in machine condition testing

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University of Technology and Life Sciences

Summary

The paper introduces the issue of fuzzy sets in the process of diagnosing the condition of machines and technical objects. It deals with a fuzzy model construction and structure in details. What is more, this paper presents the software which is ca-pable of modelling the structure of a technical object.

Keywords: structure of fuzzy model, software for fuzzy model 1. Introduction

In the process of machine operation, technical objects need to be frequently diagnosed, which is necessary for machine condition assessment through giving appropriate values of diagnostic parameters. Such parameters can be used only for the definition of measurable quantities. Each operating machine has periods of unavailability due to damages of its systems and subsystems. In such a case, a general description of the technical object, accounting for the impact of damages, is not possible. If the above-mentioned description can be achieved, then it is not possible to deter-mine it on the basis of dependences defining particular damage [1,2,7].

a) S2 b) linia decyzyjna L(S)=S1-2S2=0 L(S)= S1-3=0 L(S)<0 4 ZDATNY L(S1)<0 L(S1)>0 3 ZDATNY NIEZDATNY 2 L(S)>0 1 NIEZDATNY -1 0 1 2 3 4 5 6 S1 0 1 2 3 4 5 6 7 8 9 S1

Therefore, different diagnostic models which reflect machine simplified structure found appli-cation in diagnosing the condition of technical objects. Models for the determination of technical object condition include: analytical, neuron and fuzzy ones. This article deals with a fuzzy model of technical object condition diagnosing. The use of the model for assessment of machine condi-tion should transform diagnostic signs into technical object condicondi-tion space.

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2. Fuzzy model

Fuzzy logic, as compared to the classical one, is different in terms of values that can be reached. In classical logic only two values are available: 0 or 1, whereas fuzzy logic enables value increase within the range from 0 to 1 [e.g. 0,3;0,65; and so on]. The above-mentioned ‘fuzziness’ of boundaries found application in modelling nonlinear objects which provide uncertain and inaccurate information. The information obtained from diagnostic examination includes signals and disruptions caused by, e.g. vibrations. It also happens that a developed diagnostic model does not reflect the object actual state as such; a definition of its physical and mechanical characteristics will not be respected by designers because of their incomplete knowledge of the object. While modelling a machine condition with the use of fuzzy logic, it is necessary to combine expert knowledge of a given subject with nominal measurement data. The role of an expert is to define its structure with initial values of model parameters. Whereas, values of nominal parameters are supposed to help the designer to model properly.

Fig. 1.Exemplary structure of a fuzzy model

Source: [3].

3. Fuzzy model structure

A typical process of fuzzy inference occurs at the following stages [3]: A. Fuzzification.

B. Inference. C. Defuzzification.

At the stage of fuzzification, values x1*, x2*, represented by real numbers, are introduced on

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fuzzifi-cation is performed, that is the calculation of input affinity degree (sharp values x1*, x2*) to

particular fuzzy sets Ai, Bi.

In order to make proper fuzzy sets, affinity functions ȝAi(x1*), ȝBi(x2*) need to be precisely

defined qualitatively (application of a proper function) and quantitatively (application of the function coefficient). Both the parameters and the shape of the affinity function have a large influence on model accuracy.

At the stage of inference, the so called resultant affinity function ȝAi(x1*), ȝBi(x2*) and outcome

function of membership ȝwyn(y) of the model output are calculated on the basis of input affinity

degrees. In order to carry out inferential calculations, it is necessary to define: • base of rules,

• mechanism (function) of inference, • function of model output affinity.

The base of rules contains logical rules defining cause-effect dependences existing in the sys-tem between fuzzy sets of inputs and outputs. For example, the base of rules can have the follow-ing form;

IF (xi = A1) AND (x2 = B1) THEN (y = C1) (1)

IF (xi = A1) AND (x2 = B2) THEN (y = C2)

when: A1, A2 B1, B2– fuzzy sets of inputs, Ci, C2, C3– fuzzy sets of the output.

Fuzzy inference requires the assessment of fulfilment degree (truthfulness) of premises of par-ticular rules. There is a dependency between the fulfilment degree of the premise and the share of a particular rule in the definition of the rule base resultant conclusion.

During the operation of defuzzification, while carrying out the inference process, a certain

fuzzy set is obtained. This set can be given some linguistic meaning. However, in certain situations it is desirable to obtain numerical values in the inference result.

The defuzzification operation enables a transformation of a random fuzzy set A, described in space Y into a certain numerical value

y

₀

∈

Y

Among methods used for defuzzification, the following can be distinguished [4]:

• The gravity centre method – this method involves determining the gravity centre of a figure obtained under the function of affinity:

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• The method of gravity centre indexation – used when affinity function fragments with values lower than the parameter are to be eliminated

_α

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y dy

y

y dy

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∈

≥

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p(S/X ) p S n ds Sgr ( / ) −∞

³

p S z ds Sg r ( / ) ∞

³

p(S/z) p (S/n) Z DAT N Y NIEZD AT NY symptom stanu X

• The maximum method – for this method, numerical value (after defuzzification ) is chosen from a set of affinity function argument values for which maximum value is accepted. • The method of height – in this case, the system input value is obtained from inference results

for each of the rules of the type if-then. If the location of the gravity centre for a fuzzy set A is the result of the inference for the i – the rule, it is determined as y, and the maximum value of affinity function A’ as

τ

_i

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4. Database in a fuzzy model

Database which is indispensable for appropriate operation of the whole model is of big im-portance in technical diagnostics [1,2,7]. The classifier is constructed on the basis of an appropri-ate fuzzy set. Data is obtained as a result of tests called teaching tests, covering a set of objects. Conclusions resulting from an analysis of teaching tests will be generalized onto the whole set U, which means that classification rules are considered to be appropriate for set U.

Teaching classifier, determined on the basis of approximate criteria, can be denoted in a form of a classifier. Each element ck ck*( u) of a fuzzy teaching classifier c*( u) is a value of affinity

function of the k – the class of set K:

(5) where the values of the affinity function are interpreted in the following way:

• ck*( u)=1 when object u has all the attributes characteristic of objects belonging to the k –

the class (total affinity to the class);

{ }

(

)

(

)

1 0 ₁ 1

'

I i i I _i I i i i i

y

HM A

τ α

= = =

−

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• 0<ck*( u) <1 when the object u does not have any attributes characteristic of objects

be-longing to the k – the class (total lack of object affinity to a class);

• when the object u has partial attributes characteristic of objects belonging to the k – the class (partial lack of object affinity to a class).

Fuzzy classifier makes possible an approximate description of the object. This description is to provide a basis for making a decision concerning e.g. a possibility or its lack of machine further operating. In order to obtain a non-fuzzy classifier, called a sharp classifier, different sharpening operators are applied [4,7].

μ B A μ B A μ B A

1 1 1

x 1 x 2 x 3

Δ B / Δ A Δ B / Δ A Δ B / Δ A

5. Fuzzy model system

For a description of relations between the machine input and output system[2] is used. How-ever, in this case, having images of inputs and outputs, we have to do with machine system model. Such a system is described by a set of momentary values of parameters. The machine model condition in time t does not depend on later inputs; that is, it is defined by earlier inputs of chine model and by its initial state. An important assumption that must be accounted for in ma-chine system model is the acceptance of cause-effect approach.

However, such an assumption requires caution due to the differences between the real object and the model. Thus, the state of machine model in time t provides information on earlier inputs and outputs of machine model, indispensable for the definition of machine output in time t. 6. Object structure modelling in MATLAB software

MATLAB software is a program supporting an engineer’s job in the field of mechanics, elec-tronics, mathematics. This program contains a set of mathematical functions necessary for particu-lar calculation stages, as well as a possibility of generation of graphic charts [5]. Also, the above-mentioned program contains many additional libraries and applications called toolboxes. Applica-tions are to support the design of systems, models, and controllers.

Application of Fuzzy Logic Toolbox provides complete environment for the creation of mod-els of dynamic systems with the use of fuzzy sets and fuzzy inference rules, as well as tools for designing intelligent control systems, whose operation is based on elements of fuzzy logic. This application aims at supporting a designer at the stage of a fuzzy model design [6].

An example can be modelling a structure of an object consisting of 2 input parameters and one output. Input parameters account for the impact of the crankshaft rotational speed value (CRSV)

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and the driver driving style (DS), and as the model output a unit fuel consumption (FC) by a theoretical vehicle was accepted.

An inference system consistent with the following rules was formulated:

1. If [CSRV)= neutral gear revolutions ] and [DS = quiet] then [FC = increased] 2. If [CSRV = medium] and DS = quiet] then [DS = nominal]

3. If [CSRV = max] and [DS = quiet] then [FC = increased]

4. If [ CRSV= neutral gear revolution] and [DS = dynamic] then [FC = nominal] 5. If [CRSV = medium] and [DS = dynamic] then [FC = increased]

6. If [CSRV = max] and [DS = dynamic] then [FC = significantly increased] For a driving style it was accepted:

– quiet with values (0;0,55) –dynamic with values (0,45;1).

For rotational speed (WK) it was accepted: – neutral gear rev with values (0;0,35) – medium with values (0,35,0,85) – maximal with values (0,85;1). For fuel consumption it was accepted: – nominal with values (0;0,4)

– increased with values (0,3;0,7)

– significantly increased with values (0,6;1)

Fig. 2. 3D chart of unit fuel consumption in the function of rotational speed of the camshaft and the style of driving

In the chart above it can be read that for different values of input parameters the result of out-put parameter is diversified.

Particular input states are fuzzy field at the model input, which results in their being assigned by an inappropriate classifier. Classifiers were selected in the range [0,1], so as to make it possible to program input states in terms of fuzzy logic.

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In the chart above it can be read that for the mean rotational speed of the crankshaft fuel con-sumption for quiet style of driving is of nominal value.

When input values are changed, fuel consumption values change as well. In extreme loca-tions with maximum revoluloca-tions of the crankshaft and dynamic style of driving fuel consumption increases to maximum values ( to value close to one).

7. Conclusions

Fuzzy systems are automats which use fuzzy logic rules in order to make a decision in uncer-tain conditions. Such an automat provides a ceruncer-tain base of knowledge and inference rules, and after observing the environment and the process of inference it makes a decision. The base and the rules of inference come from an expert who creates the system. This, its efficiency largely depends on the knowledge of experts and their ability to model it by means of fuzzy logic. A fuzzy model is used more and more often in objects that can be found everywhere.

An example can be a traffic light which changes colours of lights depending on traffic inten-sity or different kinds of washing machine programmers. An advantage of the application of fuzzy models is their structure of nonlinear character, which has a large influence on their application in machine and engineering object operating.

Bibliography

1. Cholewa W.: Diagnostyka techniczna maszyn, Skrypty PS., Gliwice 1992.

2. Cholewa W.: Metoda diagnozowania maszyn z zastosowaniem zbiorów rozmytych, Roz-prawa habilitacyjna, Gliwice 1983.

3. Praca zbiorowa pod redakcją Korbicza J.: Diagnostyka procesów, WNT, Warszawa 2002. 4. ŁĊski J.: Systemy neuronowo-rozmyte, WNT, Warszawa 2008.

5. Mrozek B.: Matlab i Simulink. Poradnik uĪytkownika, Wydawnictwo Helion, Gliwice 2004. 6. http://we.pb.edu.pl/~kaie/kaie-md/MSI/Opis_funkcji_Fuzzy_tlbx.pdf

7. ĩółtowski B.: Projektowanie eksperymentów w diagnostyce maszyn. Seria rozprawy.Nr.1. 8. WSOWRiA. ToruĔ. 1984.

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ZBIORY ROZMYTE JAKO ELEMENTY SZTUCZNEJ INTELIGENCJI W DIAGNOZOWANIU MASZYN

Streszczenie

Artykuł wprowadza w problematykĊ moĪliwoĞci wykorzystania zbiorów rozmy-tych w diagnozowaniu stanu obiektów technicznych. Formalizmy matematyczne uka-zują istotĊ zbiorów rozmytych wymagających wypracowania kryteriów ostrzących dla podejmowania racjonalnych decyzji. Zaproponowano tu oprogramowanie umoĪ-liwiające modelowanie stanu obiektów w warunkach niepewnoĞci, z wykorzystaniem zbirów rozmytych.

Słowa kluczowe: niepewnoĞci, rozmytoĞü decyzji, eksploatacja, diagnostyka, uszkodzenia, stan techniczny

This paper is a part of WND-POIG.01.03.01-00-212/09 project.

Gabriel GajdziĔski Bogdan ĩółtowski

University of Technology and Life Sciences Faculty of Mechanical Engineering